Statistical Optimality and Computational Efficiency of Nystrom Kernel PCA

Kernel methods provide an elegant framework for developing nonlinear learning algorithms from simple linear methods. Though these methods have superior empirical performance in several real data applications, their usefulness is inhibited by the significant computational burden incurred in large sample situations. Various approximation schemes have been proposed in the literature to alleviate these computational issues, and the approximate kernel machines are shown to retain the empirical performance. However, the theoretical properties of these approximate kernel machines are less well understood. In this work, we theoretically study the trade-off between computational complexity and statistical ac-curacy in Nystro center dot m approximate kernel principal component analysis (KPCA), wherein we show that the Nystro center dot m approximate KPCA matches the statistical performance of (non -approximate) KPCA while remaining computationally beneficial. Additionally, we show that Nystro center dot m approximate KPCA outperforms the statistical behavior of another popular approximation scheme, the random feature approximation, when applied to KPCA.

Automated summary

In this work, the trade-off between computational complexity and statistical accuracy in Nystro approximate kernel principal component analysis (KPCA) is theoretically studied. It is shown that Nystro approximate KPCA matches the statistical performance of (non-approximate) KPCA while remaining computationally beneficial. Additionally, Nystro approximate KPCA outperforms the statistical behavior of the random feature approximation when applied to KPCA.